Let f: X->B be a fibred surface of genus g whose general fibre is a double cover of a smooth curve of genus gamma. We show that, for g > 4gamma+1, the number 4(g-1)/(g-gamma) is a sharp lower bound for the slope of f, proving a conjecture of Barja. Moreover, we give a characterisation of the fibred surfaces that reach the bound. In the case g = 4gamma+1 we obtain the same sharp bound under the assumption that the involutions on the general fibres glue to a global involution on X.
Let $f:S\to B$ be a finite cyclic covering fibration of a fibered surface. We study the lower bound of slope $λ_{f}$ when the relative irregularity $q_{f}$ is positive.
We study and obtain Slope inequalities for fibred irregular varieties of non-maximal Albanese dimension. We give a comparison theorem between Clifford-Severi and Slope inequalities for this type of fibrations. We also obtain a set of Slope inequalities considering the geometry of the Albanese map and the associated eventual maps.
We prove the equivalence between Clifford-Severi inequalities for good classes of varieties of maximal Albanese dimension and Slope Inequalities for fibrations of such varieties over curves. This provides a big set of new Slope Inequalities and characterizes the limit cases. It also gives a machinery to automatically obtain other higher dimensional Slope and Clifford-Severi inequalities from inequalities in low dimension. For this, we construct a continuous version of Xiao's method for irregular fibrations.
We establish the slope equality and give an upper bound of the slope for finite cyclic covering fibrations of an elliptic surface including bielliptic fibrations of genus greater than 5. We also give an upper bound of the slope for triple cyclic covering fibrations of a ruled surface and hyperelliptic fibrations, which provides a new proof of Xiao's upper bound.
We bound the slope of sweeping curves in the fourgonal locus of the moduli space of genus g algebraic curves. Our results follow from some Bogomolov-type inequalities for weakly positive rank two vector bundles on ruled surfaces.
We study slopes of finite cyclic covering fibrations of a fibered surface. We give the best possible lower bound of the slope of these fibrations. We also give the slope equality of finite cyclic covering fibrations of a ruled surface and observe the local concentration of the global signature of these surfaces on a finite number of fiber germs. We also give an upper bound of the slope of finite cyclic covering fibrations of a ruled surface.
K. Konno proved a slope equality for fibred surfaces with fibres of odd genus and general fibre of maximal gonality. More precisely he found a relation between the invariants of the fibration and certain weights of special fibres (called the Horikawa numbers). We give an alternative and more geometric proof in the case of a genus 5 fibration, under generality assumptions. In our setting we are able to prove that the fibre with positive Horikawa numbers are precisely the trigonal ones, we compute their weights explicitly and thus we exhibit explicit examples of regular surfaces with assigned in…
Let $f:\, S \to B$ be a locally non-trivial relatively minimal fibration of hyperelliptic curves of genus $g\geq 2$ with relative irregularity $q_f$. We show a sharp lower bound on the slope $λ_f$ of $f$. As a consequence, we prove a conjecture of Barja and Stoppino on the lower bound of $λ_f$ as an increasing function of $q_f$ in this case, and we also prove a conjecture of Xiao on the ampleness of the direct image of the relative canonical sheaf if $λ_f<4$.
Given a relatively minimal fibration $f: S \to \Bbb P^1$ on a rational surface $S$ with general fiber $C$ of genus $g$, we investigate under what conditions the inequality $6(g-1)\le K_f^2$ occurs, where $K_f$ is the canonical relative sheaf of $f$. We give sufficient conditions for having such inequality, depending on the genus and gonality of $C$ and the number of certain exceptional curves on $S$. We illustrate how these results can be used for constructing fibrations with the desired property. For fibrations of genus $11\le g\le 49$ we prove the inequality: $$ 6(g-1) +4 -4\sqrt g \le K_f^2…
We prove the slope inequality for a relative minimal surface fibration in positive characteristic via Xiao's approach. We also prove a better low bound for the slope of non-hyperelliptic fibrations.
Given a relatively minimal non locally trivial fibred surface f: S->B, the slope of the fibration is a numerical invariant associated to the fibration. In this paper we explore how properties of the general fibre of $f$ and global properties of S influence on the lower bound of the slope. First of all we obtain lower bounds of the slope when the general fibre is a double cover. We also obtain a lower bound depending as an increasing function on the relative irregularity of the fibration, extending previous results of Xiao. We construct several families of examples to check the assimptotical sh…
The Horikawa index and the local signature are introduced for relatively minimal fibered surfaces whose general fiber is a non-hyperelliptic curve of genus $4$ with unique trigonal structure.
We give a slope equality for fibered surfaces whose general fiber is a smooth plane curve. As a corollary, we prove a "strong" Durfee-type inequality for isolated hypersurface surface singularities, which implies Durfee's strong conjecture for such singularities with non-negative topological Euler number of the exceptional set of the minimal resolution.
In this paper we present a generalisation of a theorem due to Cornalba and Harris, which is an application of Geometric Invariant Theory to the study of invariants of fibrations. In particular, our generalisation makes it possible to treat the problem of bounding the invariants of general fibred surfaces. As a first application, we give a new proof of the slope inequality and of a bound for the invariants associated to double cover fibrations.
The existence of a Kodaira fibration, i.e., of a fibration of a compact complex surface $S$ onto a complex curve $B$ which is a differentiable but not a holomorphic bundle, forces the geographical slope $ ν(S) = c_1^2 (S) / c_2 (S)$ to lie in the interval $(2,3)$. But up to now all the known examples had slope $ ν(S) \leq 2 + 1/3$. In this paper we consider a special class of surfaces admitting two such Kodaira fibrations, and we can construct many new examples, showing in particular that there are such fibrations attaining the slope $ ν(S) = 2 + 2/3$. We are able to explicitly describe the mo…
We study from a geographical point of view fibrations of threefolds over smooth curves, such that the general fibre is of general type. We prove the non-negativity of certain relative invariants under general hypotheses and give lower bounds for the self-interssection of the relative canonical divisor of the fibration, depending on other relative invariants. We also study the influence of the relative irregularity on these bounds. A more detailed study of the lowest cases of the bounds is given.
Let M_g be the moduli space of stable curves of genus g >= 2. Let D_i be the irreducible component of the boundary of M_g such that general points of D_i correspond to stable curves with one node of type i. Let M_g^0 be the set of stable curves that have at most one node of type i>0. Let d_i be the class of D_i in Pic(M_g)_Q and h the Hodge class on M_g. In this paper, we will prove a sharp slope inequality for general stable fibrations. Namely, if $C$ is a complete curve on M_g^0, then ( (8g+4)h - g d_0 - \sum_{i=1}^{[g/2]} 4i(g-i) d_i . C ) >= 0. As an application, we can prove effective Bog…
Being inspired by Ross' construction of unstable products of certain smooth curves, we show that the product $C\times C$ of every smooth curve $C$ of genus at least 2 is not slope semistable with respect to certain polarisations. Besides, we produce examples of Kodaira-fibred surfaces of nonzero signature, which are not slope semistable with respect to some polarisations, and so they admit Kähler classes that do not contain any constant scalar curvature Kähler metrics.
This is the part II of the series under the same title. In part I, using the approach developed by Catanese--Pignatelli arXiv:math/0503294, we gave a structure theorem for hyperelliptic genus 3 fibrations all of whose fibers are 2-connected (arXiv:1209.6278 [math.AG]). In this part II, we shall give a structure theorem for smooth deformation families of these to non-hyperelliptic genus 3 fibrations. As an application, we shall give a set of sufficient conditions for our genus 3 hyperelliptic fibration above to allow deformation to non-hyperelliptic fibrations, and use this to study certain min…
Let $f:\,S \to B$ be a locally non-trivial relatively minimal fibration of genus $g\geq 2$ with relative irregularity $q_f$. It was conjectured by Barja and Stoppino that the slope $λ_f\geq \frac{4(g-1)}{g-q_f}$. We prove the conjecture when $q_f$ is small with respect to $g$; we also construct counterexamples when $g$ is odd and $q_f=(g+1)/2$.
In this paper we investigate the numerical properties of relatively minimal isotrivial fibrations $φ\colon X \lr C$, where $X$ is a smooth, projective surface and $C$ is a curve. In particular we prove that, if $g(C) \geq 1$ and $X$ is neither ruled nor isomorphic to a quasi-bundle, then $K_X^2 \leq 8 χ(\mO_X)-2$; this inequality is sharp and if equality holds then $X$ is a minimal surface of general type whose canonical model has precisely two ordinary double points as singularities. Under the further assumption that $K_X$ is ample, we obtain $K_X^2 \leq 8 χ(\mO_X)-5$ and the inequality is al…
In this paper, we first construct varieties of any dimension $n>2$ fibered over curves with low slopes. These examples violate the conjectural slope inequality of Barja and Stoppino [BS14b]. Led by their conjecture, we focus on finding the lowest possible slope when $n=3$. Based on a characteristic $p > 0$ method, we prove that the sharp lower bound of the slope of fibered $3$-folds over curves is $4/3$, and it occurs only when the general fiber is a $(1, 2)$-surface. Otherwise, the sharp lower bound is $2$. We also obtain a Cornalba-Harris-Xiao type slope inequality for families of surfaces o…
We establish sharp bounds for the slopes of curves in $\bar{M}_g$ that sweep the locus of trigonal curves, proving Stankova-Frenkel's conjectured bound of $7+6/g$ for even $g$ and obtaining the bound $7+20/(3g+1)$ for odd $g$. For even $g$, we find an explicit expression of the so-called Maroni divisor in the Picard group of the space of admissible triple covers. For odd $g$, we describe the analogous extremal effective divisor and give a similar explicit expression.
In this paper, we investigate the general notion of the slope for families of curves $f: X \to Y$. The main result is an answer to the above question when $\dim Y = 2$, and we prove a lower bound for this new slope in this case over fields of any characteristic. Both the notion and the slope inequality are compatible with the theory for $\dim Y = 0, 1$ in a very natural way, and this gives a strong evidence that the slope for an $n$-fold fibration of curves $f: X \to Y$ may be $K_{X/Y}^n / \mathrm{ch}_{n-1}(f_* ω_{X/Y})$. Rather than the usual stability methods, the whole proof of the slope in…
We focus on a class of Weierstrass elliptic threefolds that allows the base of the fibration to be a Fano surface or a numerically $K$-trivial surface. We define the notion of limit tilt stability, and show that the Fourier-Mukai transform of a slope stable torsion-free sheaf satisfying a vanishing condition in codimension 2 (e.g. a reflexive sheaf) is a limit tilt stable object. We also show that the inverse Fourier-Mukai transform of a limit tilt semistable object of nonzero fiber degree is a slope semistable torsion-free sheaf, up to modification in codimension 2.
We study the moduli of trigonal curves. We establish the exact upper bound of ${36(g+1)}/(5g+1)$ for the slope of trigonal fibrations. Here, the slope of any fibration $X\to B$ of stable curves with smooth general member is the ratio $δ_B/λ_B$ of the restrictions of the boundary class $δ$ and the Hodge class $λ$ on the moduli space $\bar{\mathfrak{M}}_g$ to the base $B$. We associate to a trigonal family $X$ a canonical rank two vector bundle $V$, and show that for Bogomolov-semistable $V$ the slope satisfies the stronger inequality ${δ_B}/{λ_B}\leq 7+{6}/{g}$. We further describe the rational…
Kodaira fibred surfaces are a remarkable example of projective classifying spaces, and there are still many intriguing open questions concerning them, especially the slope question. The topological characterization of Kodaira fibrations is emblematic of the use of topological methods in the study of moduli spaces of surfaces and higher dimensional complex algebraic varieties, and their compactifications. The paper contains some new results but is mostly a survey paper, dealing with fibrations, questions on monodromy and factorizations in the mapping class group, old and new results on Variatio…
Let f:S ->B be a relatively minimal fibred surface. In this note we give a partial affirmative answer to a conjecture of Xiao, proving that the direct image of the relative dualizing sheaf of $f$ is ample when the slope of the fibration is less than 4, if the general fibre of $f$ is non-hyperelliptic or the genus of the fibre or of the base curve is low.
It is known that the universal cover of compact Riemann surface is either the projective line, the complex plane or the unit disk. In this article we construct a very explicit family of complex surfaces that gives rise to uncountably many mutually non-biholomorphic universal covers. The slope of these surfaces, which are going to be total spaces of Kodaira fibrations, is also determined
We give a sharp lower bound for the selfintersection of a nef line bundle $L$ on an irregular variety $X$ in terms of its continuous global sections and the Albanese dimension of $X$, which we call the Generalized Clifford-Severi inequality. We also extend the result to nef vector bundles and give a slope inequality for fibred irregular varieties. As a byproduct we obtain a lower bound for the volume of irregular varieties; when $X$ is of maximal Albanese dimension the bound is $Vol(X) \geq 2 n! χω_X$ and it is sharp.
